Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.
0
votes
4answers
52 views
Evaluate a sum involving n choose r
Evaluate: $\sum_{k=0}^6 (-1)^k \binom{6}{k}$ where $\binom{n}{r}= \frac{n!}{r!(n-r)!}$.
I'm unsure how to compute the part with $\binom{6}{k}$, it should be something along the lines of ...
0
votes
0answers
59 views
Double summation including power and factorial
I am finding some trouble in computing the following sum:
$$\sum_{k=0}^\infty \frac{x^k}{k!}\;\sum_{m=0}^k\frac {y^m}{m!}$$
Could you please provide a result?
Thanks in advance
6
votes
2answers
188 views
How to find the factorial of a fraction?
From what I know, the factorial function is defined as follows:
$$n! = n(n-1)(n-2) \cdots(3)(2)(1)$$
And $0! = 1$. However, this page seems to be saying that you can take the factorial of a ...
4
votes
1answer
66 views
Last non zero digit of $n!$ [duplicate]
What is the last non zero digit of $100!$?
Is there a method to do the same for $n!$?
All I know is that we can find the number of zeroes at the end using a certain formula.However I guess that's of ...
2
votes
2answers
35 views
Simplify summation with factorial and binomial coefficients
I would like to know how to simplify the following summation:
$$\sum_{p=0}^n\quad n!\frac{(2p)!}{(p!)^2}\frac{(2(n-p))!}{((n-p)!)^2}$$
Which rules should I use to simplify it?
Thanks!
1
vote
3answers
108 views
What is $\frac{(an)!}{n!}$?
How can $\frac{(an)!}{n!}$ be expressed in terms of $a!$, $n!$, $a$, $n$ (and maybe Pochhammers)?
4
votes
3answers
168 views
Factorial of infinity
So, I've read in this article that:
$$\zeta'(0) = \log\sqrt\frac{1}{2\pi}$$
And that:
$$e^{-\zeta'(0)} = 1\cdot2\cdot3\cdot\ldots\cdot\infty = \infty! = \sqrt{2\pi}$$
I found this result very ...
4
votes
0answers
55 views
Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$
I've been told that the approach below will not work.
I would be interested if someone could help me to understand what will go wrong.
Let:
$$\psi(x) = \sum\limits_{p^k \le x} \ln p$$
So that (see ...
0
votes
1answer
38 views
Looking for suggestions on how to proceed with showing that:
for $x \ge 2863:$
$$\ln\left(\left\lfloor\frac{x}{6}\right\rfloor!\right) < \sum_{k=5}^{\infty}-\mu(k)\ln\left(\left\lfloor\frac{x}{k}\right\rfloor!\right)$$
I've written a java application which ...
5
votes
3answers
112 views
Prove that $(a+1)(a+2)…(a+b)$ is divisible by $b!$ [duplicate]
The problem is following, prove that:
$$(a+1)(a+2)...(a+b)\text{ is divisible by } b!\text{ for every positive integer a,b}$$
I've tried solving this problem using mathematical induction, but I ...
2
votes
3answers
67 views
Factorial Equality Problem
I'm stuck on this problem, any help would be appreciated.
Find all $n \in \mathbb{Z}$ which satisfy the following equation:
$${12 \choose n} = \binom{12}{n-2}$$
I have tried to put each of them ...
1
vote
1answer
61 views
Gamma function question
Gamma function is also known as generalized factorial function .
why does the term "generalized" have been used ?
Again, why is the Gamma function called Euler's second integral?
0
votes
1answer
37 views
Simple question during a proof. Reducing a factorial…
So I'm reading over a proof-review and stuck on how they managed to convert:
$\limsup \displaystyle \frac{|n!|^\frac{1}{2}}{|(n+1)!^\frac{1}{2}|} = \limsup \displaystyle\frac{1}{(n+1)^\frac{1}{2}}$
...
3
votes
2answers
83 views
A question about prime factorization of n!
Prove that for any integer $K$, There exists a natural number $N$ so that in the prime factorization of $N!$ we can find at least $K$ prime numbers which their powers are exactly 1.
4
votes
0answers
78 views
Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?
I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
0
votes
0answers
58 views
Using Stirling's Formula to approximate a difference of logarithms of factorials in the same way as Jitsuro Nagura.
In Jitsuro Nagura's classic proof of a prime existing between $x$ and $\frac{6x}{5}$, he uses Stirling's formula to show that:
$$T\left(x\right) - T\left(\frac{x}{2}\right) - ...
0
votes
0answers
42 views
Do these inequalities regarding the gamma function and factorials work?
I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In a previous question, I asked whether the following inequality is ...
2
votes
0answers
55 views
Trying to generalize an inequality from Jitsuro Nagura: Does this work?
I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$:
...
0
votes
1answer
28 views
How do you evaluate an inequality that involves logarithms of factorials?
For $x > 1$, $n > 2$ with $2 \mid x+1$ and $n \mid x+1$, does it then follow that:
$$\log(\lfloor\frac{x}{2}\rfloor!) - \log(\lfloor\frac{x}{n}\rfloor!) \le \log(\lfloor\frac{x+1}{2}\rfloor!) - ...
0
votes
0answers
31 views
Does it follow that if $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$, $\log(\lfloor\frac{x}{2}\rfloor!) \le \log\Gamma(\frac{x+1}{2})$?
The answer seems to be yes.
Here's my reasoning.
Let $\{x\} = x - \lfloor{x}\rfloor$
Assume $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$
$$\log(\lfloor\frac{x}{2}\rfloor!) = ...
1
vote
3answers
65 views
Testing for convergence in Infinite series with factorial in numerator
I have the following infinite series that I need to test for convergence/divergence:
$$\sum_{n=1}^{\infty} \frac{n!}{1 \times 3 \times 5 \times \cdots \times (2n-1)}$$
I can see that the denominator ...
4
votes
1answer
132 views
Sum involving the hypergeometric function, power and factorial functions
I am finding some trouble in calculating the following sum involving the hypergeometric function, power and factorial functions.
$$
\sum_{y=1}^\infty \mathrm{e}^z \cdot {}_1F_1\left(1-y;2;-z\right) ...
3
votes
2answers
38 views
Identity of binomial series with factorial.
I'm looking for a simple identity for the formula:
$$
\sum_{k = 0}^{p} \binom{p}{k} \cdot k! \cdot x^k
$$
In words, I have $p$ "players" who can choose to play or not (every player is represented by ...
4
votes
0answers
71 views
Inequality problem with factorials
I am not sure if this kind of "question" is welcome on MSE. Here is an olympiad-like problem that I would like to share with you:
Let $a,b,c$ be nonnegative integers. Prove that
$$
...
3
votes
2answers
30 views
How to define a factorial using multiple sets
I am currently studying a photography course, and I have run into a bit of difficulty with one of my projects in relation to combinatorics. I have a key rack and there are 39 hooks on this key rack. I ...
0
votes
1answer
104 views
Derivative of $\frac{e^x}{x!}$
I am having a bit of trouble putting all the differentiation rules together with the following problem:
$$ \frac{d}{da} \Bigg(\frac{a^x}{x!}e^{-a}\Bigg)$$
Where $x$ is a discrete variable and $a$ is ...
1
vote
1answer
65 views
Comparing rates of change: which function increases faster?
I am comparing two functions for $x \ge 1$:
$$f(x) = \ln(\lfloor\frac{x}{9}\rfloor!) - \ln(\lfloor\frac{x}{10}\rfloor!) - \ln(\lfloor\frac{x}{90}\rfloor!)$$
$$g(x) = (2.07766)\sqrt{\frac{x}{9}} + ...
1
vote
1answer
46 views
Is this a valid way to evaluate a function based on factorials? What would be a better way?
I am working on the following factorial function:
$$f(x) = [\ln(\lfloor\frac{x}{11}\rfloor!) - \ln(\lfloor\frac{x}{12}\rfloor!) - \ln(\lfloor\frac{x}{132}\rfloor!)] + ...
0
votes
1answer
68 views
Proving that a specific gamma function is a guaranteed lower bound for a factorial function
In reviewing Ramanujan's proof of Bertrand's postulate, Ramanujan observes that:
$$\ln\Gamma(x) - 2\ln\Gamma(\frac{x+1}{2}) \le \ln(\lfloor{x}\rfloor!) - 2\ln(\lfloor\frac{x}{2}\rfloor!)$$
I have ...
0
votes
1answer
37 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
1
vote
0answers
28 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
2
votes
1answer
69 views
Funny graph of $x!$ by a graphing program
I obtained a bizarre graph of $x!$, a function which I believe is only defined at positive integer domain.
What causes such an error?. It would be interesting if someone can explain the method such a ...
8
votes
1answer
205 views
Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?
I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident.
In particular, Ramanujan's does the following ...
1
vote
0answers
101 views
Using the gamma function as an upper and lower bound to the logarithm of a factorial function.
I am trying to find an upper and lower bound for the following function:
$$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$
where
...
1
vote
1answer
42 views
Understanding the upper and lower bounds of the error estimate in Stirling's Approximation
Based on the Wikipedia article on Stirling Approximation:
$n! = \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n e^{\lambda_n}$
where $\frac{1}{12n+1} < \lambda_n < \frac{1}{12n}$
How would this ...
15
votes
3answers
168 views
Proving that $\frac{(k!)!}{k!^{(k-1)!}}$ is an integer
I have to prove that:
$$\frac{(k!)!}{k!^{(k-1)!}} \in \Bbb Z$$
for any $k \geq 1, k \in \Bbb N$
Tried doing $t = k!$ which would give $$\frac{t!}{t^{t/k}}$$
But I think I just made it harder, and ...
1
vote
2answers
52 views
How to show the double factorial isn't a polynomial
$(2n-1)!! = \dfrac{(2n)!}{2^{n} \times n!}$
I was wondering how you prove the double factorial is exponential.
I guess you have to prove that for all $m$ and $\alpha$ that there exists an $n$ such ...
2
votes
2answers
144 views
Analyzing the lower bound of a logarithm of factorials using Stirling's Approximation
I am trying to get the lower bound for:
$f(x) = \ln(\lfloor\frac{x}{4}\rfloor!) - \ln(\lfloor\frac{x}{5}\rfloor!) -\ln(\lfloor\frac{x}{20}\rfloor!) - 2(1.03883)(\sqrt{\frac{x}{4}}) - ...
2
votes
2answers
44 views
Proving factorial attribute
Given that : $$ \sum_{i=1}^{k} a_i = n $$
I am asked to prove that:
$$\prod_{j=1}^{k} a_j! $$
divises $n!$
I saw that it works for $k=1$, and for $k=2$ I tried :
$$\frac{n!}{a_1! a_2!} = \frac{(a_1 ...
3
votes
1answer
55 views
Reasoning about the Chebyshev functions: How does one check an upper bound based on the second Chebyshev function?
In Ramanujan's proof of Bertrand's Postulate, Ramanujan states:
$\log([x]!) - 2\log([\frac{1}{2}x]!) \le \psi(x) - \psi(\frac{1}{2}x) + \psi(\frac{1}{3}x)$
where:
$\vartheta(x) = \sum_{p \le x} ...
8
votes
0answers
220 views
Understanding Ramanujan's approach in his proof of Bertrand's Postulate
I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$
What would be wrong with this approach for ...
2
votes
3answers
59 views
1
vote
2answers
95 views
Simplify summation of factorials
Hello I guess this equality is true but I don't know how to solve it.
$$\sum_{x=0}^{m(1-\text{sel})} (m-1-x)! (m \cdot \text{sel}) \frac{(m(1-\text{sel}))!}{(m(1-\text{sel})-x)!}(x+1) = ...
4
votes
3answers
60 views
Calculating limit involving factorials.
I want to show that $\lim\limits_{k\to\infty} \frac{\pi^kk!}{(2k+1)!} = 0$. I've been trying to use the squeeze theorem, but am having a hard time finding some expression $P$ involving $k$ that is ...
23
votes
8answers
2k views
Do factorials really grow faster than exponential functions?
Having trouble understanding this. Is there anyway to prove it?
1
vote
2answers
33 views
Constrictions on A.P with factorials.
There are five numbers $(a_1,a_2,a_3,a_4,a_5)$, such that they are in Arithmetic Progression.
Given that $a_1$ and $a_2$ are factorials, is there a possibility that either $a_4$ OR $a_5$ is a ...
7
votes
3answers
170 views
Compact formula for $\sum_k k!$ [duplicate]
Is there any compact formula for:
$$\sum_{k=0}^n k!$$
I've tried to find it using one method for summation, but I was able to receive only compact formula for $\sum_k k! \cdot k = (n+1)!-1$
I've ...
2
votes
1answer
32 views
How to perform the summation/addition of binomial coefficients?
From my textbook:
$$
\begin{align}
\sum_{k=0}^n \binom {m+k}m &= \binom {m+n}m + \sum_{k=0}^{n-1} \binom {m+k}m\\\\\\
&= \binom {m+n}m + \binom {m+n}{m+1}\\\\\\
&= \binom {m+1+n}{m+1}
...
2
votes
4answers
229 views
What are the rules for factorial manipulation?
I know that
$$(k+1)! - 1 + (k+1)(k+1)! = (k+2)! - 1$$
thanks to wolframalpha, but I don't understand the steps for simplification, and I can't seem to find any rules about factorial manipulations ...
4
votes
2answers
85 views
Series involving factorials
How would one go about proving
$$\int_{0}^1\frac{e^x-1}{x/2}\ dx=\sum_{n=0}^\infty\frac{1}{\binom{n+2}{2}}\frac{1}{n!}(0!+1!+2!+3!+...+n!)$$
